Do All Cells Have a Nucleus?
No, not all cells have a nucleus.
Cells are fundamentally divided into two main categories based on the presence or absence of a nucleus:
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Eukaryotic cells: These cells possess a true nucleus, a membrane-bound organelle containing the cell's genetic material (DNA). Examples include plant, animal, and fungal cells. The nucleus protects the DNA and separates it from the rest of the cell's contents, the cytoplasm. As stated in The Origin and Evolution of Cells, "[eukaryotic cells have a nucleus in which the genetic material is separated from the cytoplasm."]
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Prokaryotic cells: These cells lack a nucleus. Their genetic material is located in a region called the nucleoid, which isn't enclosed by a membrane. Bacteria and archaea are examples of organisms composed of prokaryotic cells. Multiple sources, including The Cell (NCBI Bookshelf) and The Cell Nucleus (Molecular Expressions), confirm that "[prokaryotic cells (bacteria) lack a nuclear envelope" and that "Simpler one-celled organisms (prokaryotes), like the bacteria and cyanobacteria, don't have a nucleus."
Exceptions: Mature Red Blood Cells
Even within eukaryotic organisms, there are exceptions. Mature red blood cells in mammals, for instance, lose their nucleus during development. This is highlighted in resources such as Red Blood Cells: Function, Role & Importance which notes that "Red blood cells don't have a nucleus like white blood cells, allowing them..." to efficiently carry oxygen. The loss of the nucleus in these cells allows for more space for hemoglobin, the oxygen-carrying protein. This is a specialized adaptation, not a universal feature of eukaryotic cells.
Summary
The presence or absence of a nucleus is a crucial factor differentiating prokaryotic and eukaryotic cells. While most eukaryotic cells contain a nucleus, exceptions exist, such as mature mammalian red blood cells.